Lesson 24: Introduction to Simultaneous Equations · Lesson 24: Introduction to Simultaneous Equations 380 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015

NYS COMMON CORE MATHEMATICS CURRICULUM 8•4 Lesson 24

Lesson 24: Introduction to Simultaneous Equations

380

This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G8-M4-TE-1.3.0-09.2015

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.


Student Outcomes

Students know that a system of linear equations, also known as simultaneous equations, is when two or more

equations are involved in the same problem and work must be completed on them simultaneously. Students

also learn the notation for simultaneous equations.

Students compare the graphs that comprise a system of linear equations in the context of constant rates to

answer questions about time and distance.

Lesson Notes

Students complete Exercises 1–3 as an introduction to simultaneous linear equations in a familiar context. Example 1

demonstrates what happens to the graph of a line when there is change in the circumstances involving time with

constant rate problems. It is in preparation for the examples that follow. It is not necessary that Example 1 be shown to

students, but it is provided as a scaffold. If Example 1 is used, consider skipping Example 3.

Classwork

Exercises 1–3 (5 minutes)

Students complete Exercises 1–3 in pairs. Once students are finished, continue with the Discussion about systems of

linear equations.

Exercises

1. Derek scored 𝟑𝟎 points in the basketball game he played, and not once did he go to the free throw line. That means

that Derek scored two-point shots and three-point shots. List as many combinations of two- and three-pointers as

you can that would total 𝟑𝟎 points.

Number of Two-Pointers Number of Three-Pointers

𝟏𝟓 𝟎

𝟎 𝟏𝟎

𝟏𝟐 𝟐

𝟗 𝟒

𝟔 𝟔

𝟑 𝟖

Write an equation to describe the data.

Let 𝒙 represent the number of 𝟐-pointers and 𝒚 represent the number of 𝟑-pointers.

𝟑𝟎 = 𝟐𝒙 + 𝟑𝒚

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US






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2. Derek tells you that the number of two-point shots that he made is five more than the number of three-point shots.

How many combinations can you come up with that fit this scenario? (Don’t worry about the total number of

points.)

Number of Two-Pointers Number of Three-Pointers

𝟔 𝟏

𝟕 𝟐

𝟖 𝟑

𝟗 𝟒

𝟏𝟎 𝟓

𝟏𝟏 𝟔

Write an equation to describe the data.

Let 𝒙 represent the number of two-pointers and 𝒚 represent the number of three-pointers.

𝒙 = 𝟓 + 𝒚

3. Which pair of numbers from your table in Exercise 2 would show Derek’s actual score of 𝟑𝟎 points?

The pair 𝟗 and 𝟒 would show Derek’s actual score of 𝟑𝟎 points.

Discussion (5 minutes)

There are situations where we need to work with two linear equations simultaneously. Hence, the phrase

simultaneous linear equations. Sometimes a pair of linear equations is referred to as a system of linear

equations.

The situation with Derek can be represented as a system of linear equations.

Let 𝑥 represent the number of two-pointers and 𝑦 represent the number of three-pointers; then

{2𝑥 + 3𝑦 = 30𝑥 = 5 + 𝑦.

The notation for simultaneous linear equations lets us know that we are looking for the ordered pair (𝑥, 𝑦)

that makes both equations true. That point is called the solution to the system.

Just like equations in one variable, some systems of equations have exactly one solution, no solution, or

infinitely many solutions. This is a topic for later lessons.

Ultimately, our goal is to determine the exact location on the coordinate plane where the graphs of the two

linear equations intersect, giving us the ordered pair (𝑥, 𝑦) that is the solution to the system of equations.

This, too, is a topic for a later lesson.







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We can graph both equations on the same coordinate plane.

Because we are graphing two distinct lines on the same graph, we identify the lines in some manner. In this

case, we identify them by the equations.

Note the point of intersection. Does it satisfy both equations in the system?

The point of intersection of the two lines is (9, 4).

2(9) + 3(4) = 30

18 + 12 = 30

30 = 30

9 = 4 + 5

9 = 9

Yes, 𝑥 = 9 and 𝑦 = 4 satisfies both equations of the system.

Therefore, Derek made 9 two-point shots and 4 three-point shots.







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Example 1 (6 minutes)

Pia types at a constant rate of 3 pages every 15 minutes. Pia’s rate is 1

5 pages per minute. If she types 𝑦 pages

in 𝑥 minutes at that constant rate, then 𝑦 =15

𝑥. Pia’s rate represents the proportional relationship 𝑦 =15

𝑥.

The following table displays the number of pages Pia typed at the end of certain time intervals.

Number of Minutes (𝒙) Pages Typed (𝒚)

0 0

5 1

10 2

15 3

20 4

25 5

The following is the graph of 𝑦 =15

𝑥.

Pia typically begins work at 8:00 a.m. every day. On our graph, her start time is reflected as the origin of the

graph (0, 0), that is, zero minutes worked and zero pages typed. For some reason, she started working

5 minutes earlier today. How can we reflect the extra 5 minutes she worked on our graph?

The 𝑥-axis represents the time worked, so we need to do something on the 𝑥-axis to reflect the

additional 5 minutes of work.







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If we translate the graph of 𝑦 =15

𝑥 to the right 5 units to reflect the additional 5 minutes of work, then we

have the following graph:

Does a translation of 5 units to the right reflect her working an additional 5 minutes?

No. It makes it look like she got nothing done the first 5 minutes she was at work.

Let’s see what happens when we translate 5 units to the left.

Does a translation of 5 units to the left reflect her working an additional 5 minutes?

Yes. It shows that she was able to type 1 page by the time she normally begins work.

What is the equation that represents the graph of the translated line?

The equation that represents the graph of the red line is 𝑦 =15

𝑥 + 1.

Note again that even though the graph has been translated, the slope of the line is still equal to the constant

rate of typing.







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When we factor out 1

5 from

1

5𝑥 + 1 to give us 𝑦 =

15

(𝑥 + 5), we can better see the additional 5 minutes of

work time. Pia typed for an additional 5 minutes, so it makes sense that we are adding 5 to the number of

minutes, 𝑥, that she types. However, on the graph, we translated 5 units to the left of zero. How can we make

sense of that?

Since her normal start time was 8:00 a.m., then 5 minutes before 8:00 a.m. is 5 minutes less than 8:00

a.m., which means we would need to go to the left of 8:00 a.m. (in this case, the origin of the graph) to

mark her start time.

If Pia started work 20 minutes early, what equation would represent the number of pages she could type in 𝑥

minutes?

The equation that represents the graph of the red line is 𝑦 =15

(𝑥 + 20).


Now we will look at an example of a situation that requires simultaneous linear equations.

Sandy and Charlie run at constant speeds. Sandy runs from their school to the train station in 15 minutes, and

Charlie runs the same distance in 10 minutes. Charlie starts 4 minutes after Sandy left the school. Can Charlie

catch up to Sandy? The distance between the school and the station is 2 miles.

What is Sandy’s average speed in 15 minutes? Explain.

Sandy’s average speed in 15 minutes is 2

15 miles per minute because she runs 2 miles in 15 minutes.

Since we know Sandy runs at a constant speed, then her constant speed is 2

15 miles per minute.

What is Charlie’s average speed in 10 minutes? Explain.

Charlie’s average speed in 10 minutes is 2

10 miles per minute, which is the same as

1

5 miles per minute

because he walks runs 1 mile in 5 minutes.

Since we know Charlie runs at a constant speed, then his constant speed is 1

5 miles per minute.

Suppose Charlie ran 𝑦 miles in 𝑥 minutes at that constant speed; then 𝑦 =15

𝑥. Charlie’s speed represents the

proportional relationship 𝑦 =15

𝑥.

Let’s put some information about Charlie’s run in a table:

Number of Minutes (𝒙) Miles Run (𝒚)

0 0

5 1

10 2

15 3

20 4

25 5

MP.2







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At 𝑥 minutes, Sandy has run 4 minutes longer than Charlie. Then the distance that Sandy ran in 𝑥 + 4 minutes

is 𝑦 miles. Then the linear equation that represents Sandy’s motion is

𝑦

𝑥 + 4=

2

15

𝑦 =2

15(𝑥 + 4)

𝑦 =2

15𝑥 +

8

15.

How many miles had Sandy run by the time Charlie left the school?

At zero minutes in Charlie’s run, Sandy will have run 8

15 miles.

Let’s put some information about Sandy’s run in a table:

Number of Minutes (𝒙) Miles Run (𝒚)

5 18

15= 1

3

15= 1

1

5

10 28

15= 1

13

15

15 38

15= 2

8

15

20 48

15= 3

3

15= 3

1

5

25 58

15= 3

13

15

Now let’s sketch the graphs of each linear equation on a coordinate plane.

MP.2







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A couple of comments about our graph:

- The 𝑦-intercept point of the graph of Sandy’s run shows the exact distance she has run at the moment

that Charlie starts walking. Notice that the 𝑥-intercept point of the graph of Sandy’s run shows that

she starts running 4 minutes before Charlie.

- Since the 𝑦-axis represents the distance traveled, the point of intersection of the graphs of the two

lines represents the moment they have both traveled the same distance.

Recall the original question that was asked: Can Charlie catch up to Sandy? Keep in mind that the train station

is 2 miles from the school.

It looks like the lines intersect at a point between 1.5 and 2 miles; therefore, the answer is yes, Charlie

can catch up to Sandy.

At approximately what point do the graphs of the lines intersect?

The lines intersect at approximately (10, 1.8).

A couple of comments about our equations 𝑦 =15

𝑥 and 𝑦 =2

15𝑥 +

815

:

- Notice that 𝑥 (the representation of time) is the same in both equations.

- Notice that we are interested in finding out when 𝑦 = 𝑦 because this is when the distance traveled by

both Sandy and Charlie is the same (i.e., when Charlie catches up to Sandy).

- We write the pair of simultaneous linear equations as

{𝑦 =

2

15𝑥 +

8

15

𝑦 =1

5𝑥

.


Randi and Craig ride their bikes at constant speeds. It takes Randi 25 minutes to bike 4 miles. Craig can bike 4

miles in 32 minutes. If Randi gives Craig a 20-minute head start, about how long will it take Randi to catch up

to Craig?

We want to express the information about Randi and Craig in terms of a system of linear equations. Write the

linear equations that represent their constant speeds.

Randi’s rate is 4

25 miles per minute. If she bikes 𝑦 miles in 𝑥 minutes at that constant rate, then

𝑦 =4

25𝑥. Randi’s rate represents the proportional relationship 𝑦 =

425

𝑥.

Craig’s rate is 4

32 miles per minute, which is equivalent to

1

8 miles per minute. If he bikes 𝑦 miles in 𝑥

minutes at that constant rate, then 𝑦 =18

𝑥. Craig’s rate represents the proportional relationship

𝑦 =18

𝑥.







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We want to account for the head start that Craig is getting. Since Craig gets a head start of 20 minutes, we

need to add that time to his total number of minutes traveled:

𝑦 =1

8(𝑥 + 20)

𝑦 =1

8𝑥 +

20

8

𝑦 =1

8𝑥 +

5

2

The system of linear equations that represents this situation is

{𝑦 =

1

8𝑥 +

5

2

𝑦 =4

25𝑥

.

Now we can sketch the graphs of the system of equations on a coordinate plane.

Notice again that the 𝑦-intercept point of Craig’s graph shows the distance that Craig was able to travel at the

moment Randi began biking. Also notice that the 𝑥-intercept point of Craig’s graph shows us that he started

biking 20 minutes before Randi.

Now, answer the question: About how long will it take Randi to catch up to Craig? We can give two answers:

one in terms of time and the other in terms of distance. What are those answers?

It will take Randi about 70 minutes or about 11 miles to catch up to Craig.

At approximately what point do the graphs of the lines intersect?

The lines intersect at approximately (70, 11).







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Exercises 4–5 (7 minutes)

Students complete Exercises 4–5 individually or in pairs.

4. Efrain and Fernie are on a road trip. Each of them drives at a constant speed. Efrain is a safe driver and travels 𝟒𝟓

miles per hour for the entire trip. Fernie is not such a safe driver. He drives 𝟕𝟎 miles per hour throughout the trip.

Fernie and Efrain left from the same location, but Efrain left at 8:00 a.m., and Fernie left at 11:00 a.m. Assuming

they take the same route, will Fernie ever catch up to Efrain? If so, approximately when?

a. Write the linear equation that represents Efrain’s constant speed. Make sure to include in your equation the

extra time that Efrain was able to travel.

Efrain’s rate is 𝟒𝟓

𝟏 miles per hour, which is the same as 𝟒𝟓 miles per hour. If he drives 𝒚 miles in 𝒙 hours at

that constant rate, then 𝒚 = 𝟒𝟓𝒙. To account for his additional 𝟑 hours of driving time that Efrain gets, we

write the equation 𝒚 = 𝟒𝟓(𝒙 + 𝟑).

𝒚 = 𝟒𝟓𝒙 + 𝟏𝟑𝟓

b. Write the linear equation that represents Fernie’s constant speed.

Fernie’s rate is 𝟕𝟎

𝟏 miles per hour, which is the same as 𝟕𝟎 miles per hour. If he drives 𝒚 miles in 𝒙 hours at

that constant rate, then 𝒚 = 𝟕𝟎𝒙.

c. Write the system of linear equations that represents this situation.

{𝒚 = 𝟒𝟓𝒙 + 𝟏𝟑𝟓𝒚 = 𝟕𝟎𝒙

d. Sketch the graphs of the two linear equations.

e. Will Fernie ever catch up to Efrain? If so, approximately when?

Yes, Fernie will catch up to Efrain after about 𝟒𝟏𝟐

hours of driving or after traveling about 𝟑𝟐𝟓 miles.







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f. At approximately what point do the graphs of the lines intersect?

The lines intersect at approximately (𝟒. 𝟓, 𝟑𝟐𝟓).

5. Jessica and Karl run at constant speeds. Jessica can run 𝟑 miles in 𝟐𝟒 minutes. Karl can run 𝟐 miles in 𝟏𝟒 minutes.

They decide to race each other. As soon as the race begins, Karl trips and takes 𝟐 minutes to recover.

a. Write the linear equation that represents Jessica’s constant speed. Make sure to include in your equation the

extra time that Jessica was able to run.

Jessica’s rate is 𝟑

𝟐𝟒 miles per minute, which is equivalent to

𝟏

𝟖 miles per minute. If Jessica runs 𝒚 miles 𝒙

minutes at that constant speed, then 𝒚 =𝟏𝟖

𝒙. To account for her additional 𝟐 minute of running that Jessica

gets, we write the equation

𝒚 =𝟏

𝟖(𝒙 + 𝟐)

𝒚 =𝟏

𝟖𝒙 +

𝟏

𝟒

b. Write the linear equation that represents Karl’s constant speed.

Karl’s rate is 𝟐

𝟏𝟒 miles per minute, which is the same as

𝟏

𝟕 miles per minute. If Karl runs 𝒚 miles in 𝒙 minutes at

that constant speed, then 𝒚 =𝟏𝟕

𝒙.


{𝒚 =

𝟏

𝟖𝒙 +

𝟏

𝟖

𝒚 =𝟏

𝟕𝒙

d. Sketch the graphs of the two linear equations.







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e. Use the graph to answer the questions below.

i. If Jessica and Karl raced for 𝟑 miles, who would win? Explain.

If the race were 𝟑 miles, then Karl would win. It only takes Karl 𝟐𝟏 minutes to run 𝟑 miles, but it takes

Jessica 𝟐𝟒 minutes to run the distance of 𝟑 miles.

ii. At approximately what point would Jessica and Karl be tied? Explain.

Jessica and Karl would be tied after about 𝟒 minutes or a distance of 𝟏 mile. That is where the graphs

of the lines intersect.

Closing (4 minutes)

Summarize, or ask students to summarize, the main points from the lesson:

We know that some situations require two linear equations. In those cases, we have what is called a system of

linear equations or simultaneous linear equations.

The solution to a system of linear equations, similar to a linear equation, is all of the points that make the

equations true.

We can recognize a system of equations by the notation used, for example: {𝑦 =

18

𝑥 +52

𝑦 =4

25𝑥

.

Exit Ticket (5 minutes)

Lesson Summary

A system of linear equations is a set of two or more linear equations. When graphing a pair of linear equations in

two variables, both equations in the system are graphed on the same coordinate plane.

A solution to a system of two linear equations in two variables is an ordered pair of numbers that is a solution to

both equations. For example, the solution to the system of linear equations {𝒙 + 𝒚 = 𝟔𝒙 − 𝒚 = 𝟒

is the ordered pair (𝟓, 𝟏)

because substituting 𝟓 in for 𝒙 and 𝟏 in for 𝒚 results in two true equations: 𝟓 + 𝟏 = 𝟔 and 𝟓 − 𝟏 = 𝟒.

Systems of linear equations are notated using brackets to group the equations, for example: {𝒚 =

𝟏𝟖

𝒙 +𝟓𝟐

𝒚 =𝟒

𝟐𝟓𝒙

.







392



Name Date


Exit Ticket

Darnell and Hector ride their bikes at constant speeds. Darnell leaves Hector’s house to bike home. He can bike the 8

miles in 32 minutes. Five minutes after Darnell leaves, Hector realizes that Darnell left his phone. Hector rides to catch

up. He can ride to Darnell’s house in 24 minutes. Assuming they bike the same path, will Hector catch up to Darnell

before he gets home?

a. Write the linear equation that represents Darnell’s constant speed.

b. Write the linear equation that represents Hector’s constant speed. Make sure to take into account that Hector

left after Darnell.








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d. Sketch the graphs of the two equations.

e. Will Hector catch up to Darnell before he gets home? If so, approximately when?








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Exit Ticket Sample Solutions

Darnell and Hector ride their bikes at constant speeds. Darnell leaves Hector’s house to bike home. He can bike the 𝟖

miles in 𝟑𝟐 minutes. Five minutes after Darnell leaves, Hector realizes that Darnell left his phone. Hector rides to catch

up. He can ride to Darnell’s house in 𝟐𝟒 minutes. Assuming they bike the same path, will Hector catch up to Darnell

before he gets home?

a. Write the linear equation that represents Darnell’s constant speed.

Darnell’s rate is 𝟏

𝟒 miles per minute. If he bikes 𝒚 miles in 𝒙 minutes at that constant speed, then 𝒚 =

𝟏𝟒

𝒙.

b. Write the linear equation that represents Hector’s constant speed. Make sure to take into account that

Hector left after Darnell.

Hector’s rate is 𝟏

𝟑 miles per minute. If he bikes 𝒚 miles in 𝒙 minutes, then 𝒚 =

𝟏𝟑

𝒙. To account for the extra

time Darnell has to bike, we write the equation

𝒚 =𝟏

𝟑(𝒙 − 𝟓)

𝒚 =𝟏

𝟑𝒙 −

𝟓

𝟑.


{𝒚 =

𝟏

𝟒𝒙

𝒚 =𝟏

𝟑𝒙 −

𝟓

𝟑


e. Will Hector catch up to Darnell before he gets home? If so, approximately when?

Hector will catch up 𝟐𝟎 minutes after Darnell left his house (or 𝟏𝟓 minutes of biking by Hector) or

approximately 𝟓 miles.


The lines intersect at approximately (𝟐𝟎, 𝟓).







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Problem Set Sample Solutions

1. Jeremy and Gerardo run at constant speeds. Jeremy can run 𝟏 mile in 𝟖 minutes, and Gerardo can run 𝟑 miles in 𝟑𝟑

minutes. Jeremy started running 𝟏𝟎 minutes after Gerardo. Assuming they run the same path, when will Jeremy

catch up to Gerardo?

a. Write the linear equation that represents Jeremy’s constant speed.

Jeremy’s rate is 𝟏

𝟖 miles per minute. If he runs 𝒚 miles in 𝒙 minutes, then 𝒚 =

𝟏𝟖

𝒙.

b. Write the linear equation that represents Gerardo’s constant speed. Make sure to include in your equation

the extra time that Gerardo was able to run.

Gerardo’s rate is 𝟑

𝟑𝟑 miles per minute, which is the same as

𝟏

𝟏𝟏 miles per minute. If he runs 𝒚 miles in 𝒙

minutes, then 𝒚 =𝟏

𝟏𝟏𝒙. To account for the extra time that Gerardo gets to run, we write the equation

𝒚 =𝟏

𝟏𝟏(𝒙 + 𝟏𝟎)

𝒚 =𝟏

𝟏𝟏𝒙 +

𝟏𝟎

𝟏𝟏


{𝒚 =

𝟏

𝟖𝒙

𝒚 =𝟏

𝟏𝟏𝒙 +

𝟏𝟎

𝟏𝟏


e. Will Jeremy ever catch up to Gerardo? If so, approximately when?

Yes, Jeremy will catch up to Gerardo after about 𝟐𝟒 minutes or about 𝟑 miles.


The lines intersect at approximately (𝟐𝟒, 𝟑).







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2. Two cars drive from town A to town B at constant speeds. The blue car travels 𝟐𝟓 miles per hour, and the red car

travels 𝟔𝟎 miles per hour. The blue car leaves at 9:30 a.m., and the red car leaves at noon. The distance between

the two towns is 𝟏𝟓𝟎 miles.

a. Who will get there first? Write and graph the system of linear equations that represents this situation.

The linear equation that represents the distance traveled by the blue car is 𝒚 = 𝟐𝟓(𝒙 + 𝟐. 𝟓), which is the

same as 𝒚 = 𝟐𝟓𝒙 + 𝟔𝟐. 𝟓. The linear equation that represents the distance traveled by the red car is

𝒚 = 𝟔𝟎𝒙. The system of linear equations that represents this situation is

{𝒚 = 𝟐𝟓𝒙 + 𝟔𝟐. 𝟓𝒚 = 𝟔𝟎𝒙

.

The red car will get to town B first.

b. At approximately what point do the graphs of the lines intersect?

The lines intersect at approximately (𝟏. 𝟖, 𝟏𝟏𝟎).





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Lesson 24: Introduction to Simultaneous Equations · Lesson 24: Introduction to Simultaneous Equations 380 This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015